To determine which statement is true regarding the similarity and volume comparison of the two wedge polyhedrons (Figures A and B), we need to recall that if two three-dimensional shapes are similar, the ratio of their volumes is equal to the cube of the ratio of their corresponding linear dimensions.
Here's how to interpret the potential volume ratios given in the options:
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If the volume ratio of two similar figures is 125:1, it implies that the ratio of their corresponding lengths is \( \sqrt[3]{125}:1 \) = \( 5:1 \). This suggests that Figure A could be $5$ times larger than Figure B linearly.
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If the volume ratio of two similar figures is 5:1, it implies that the ratio of their corresponding lengths is \( \sqrt[3]{5}:1 \). This indicates that they might not necessarily be similar in the same way since their volume ratio suggests a different scale of similarity.
Given the options provided:
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First two options (125:1) confirm the figures are similar because the volumes have a ratio of 125:1, which means they can be similar in shape with a corresponding scale factor of 5.
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The third option states they are not similar due to lack of scale factor, which is incorrect; the volume ratios imply similarity.
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The fourth option (5:1) suggests a different level of similarity and doesn't align with the concept of volume comparison directly.
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The last option states they are not similar because the volumes are not the same, which is also incorrect since volume differences do not mean lack of similarity.
Based on these evaluations, the correct statement is:
The two figures are similar because the volumes have a ratio of 125:1.
1 answer